getwd()
## [1] "/Users/rene/Desktop/CSULB MS/Fall 2020/STAT 510/Project"
# Setup
library(GGally)
## Loading required package: ggplot2
## Registered S3 method overwritten by 'GGally':
##   method from   
##   +.gg   ggplot2
library(leaps)
library(MASS)
dat = read.csv('2019b.csv', header = TRUE)
head(dat)
##   Score GDP.per.capita Social.support Healthy.life.expectancy
## 1 7.769          1.340          1.587                   0.986
## 2 7.600          1.383          1.573                   0.996
## 3 7.554          1.488          1.582                   1.028
## 4 7.494          1.380          1.624                   1.026
## 5 7.488          1.396          1.522                   0.999
## 6 7.480          1.452          1.526                   1.052
##   Freedom.to.make.life.choices Generosity Perceptions.of.corruption
## 1                        0.596      0.153                     0.393
## 2                        0.592      0.252                     0.410
## 3                        0.603      0.271                     0.341
## 4                        0.591      0.354                     0.118
## 5                        0.557      0.322                     0.298
## 6                        0.572      0.263                     0.343
y = dat$Score
x1 = dat$GDP.per.capita
x2 = dat$Social.support
x3 = dat$Healthy.life.expectancy
x4 = dat$Freedom.to.make.life.choices
x5 = dat$Generosity
x6 = dat$Perceptions.of.corruption
n = nrow(dat)
# Scatter-plot of the data
ggpairs(dat, cardinality_threshold = NULL)

ggpairs(dat, cardinality_threshold = 156)

# Step-wise Regression
# a)F-Test:
mod0 = lm(y~1)
add1(mod0, ~.+x1+x2+x3+x4+x5+x6, test='F')
## Single term additions
## 
## Model:
## y ~ 1
##        Df Sum of Sq     RSS      AIC  F value    Pr(>F)    
## <none>              192.051   34.433                       
## x1      1   121.040  71.011 -118.776 262.4976 < 2.2e-16 ***
## x2      1   115.964  76.087 -108.005 234.7110 < 2.2e-16 ***
## x3      1   116.809  75.242 -109.747 239.0755 < 2.2e-16 ***
## x4      1    61.686 130.365  -24.005  72.8697 1.238e-14 ***
## x5      1     1.104 190.946   35.533   0.8905    0.3468    
## x6      1    28.557 163.493   11.319  26.8993 6.654e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
mod1 = update(mod0, ~.+x1)
add1(mod1,~.+x2+x3+x4+x5+x6, test='F')
## Single term additions
## 
## Model:
## y ~ x1
##        Df Sum of Sq    RSS     AIC F value    Pr(>F)    
## <none>              71.011 -118.78                      
## x2      1   14.1076 56.903 -151.33 37.9322 6.177e-09 ***
## x3      1    8.6493 62.361 -137.04 21.2205 8.562e-06 ***
## x4      1   15.8450 55.166 -156.16 43.9454 5.455e-10 ***
## x5      1    3.7379 67.273 -125.21  8.5011  0.004083 ** 
## x6      1    4.6385 66.372 -127.31 10.6927  0.001329 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
mod2 = update(mod1,~.+x4)
summary(mod2)
## 
## Call:
## lm(formula = y ~ x1 + x4)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.3263 -0.4189  0.1303  0.3823  1.1176 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   2.7503     0.1546  17.789  < 2e-16 ***
## x1            1.8894     0.1308  14.442  < 2e-16 ***
## x4            2.4113     0.3637   6.629 5.45e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6005 on 153 degrees of freedom
## Multiple R-squared:  0.7128, Adjusted R-squared:  0.709 
## F-statistic: 189.8 on 2 and 153 DF,  p-value: < 2.2e-16
add1(mod2,~.+x2+x3+x5+x6, test='F')
## Single term additions
## 
## Model:
## y ~ x1 + x4
##        Df Sum of Sq    RSS     AIC F value    Pr(>F)    
## <none>              55.166 -156.16                      
## x2      1    7.8369 47.329 -178.07 25.1687 1.449e-06 ***
## x3      1    5.7092 49.457 -171.21 17.5465 4.735e-05 ***
## x5      1    0.4566 54.709 -155.46  1.2686    0.2618    
## x6      1    0.5451 54.621 -155.71  1.5170    0.2200    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
mod3 = update(mod2,~.+x2)
summary(mod3)
## 
## Call:
## lm(formula = y ~ x1 + x4 + x2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.1384 -0.4126  0.0811  0.3958  1.1332 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   2.0492     0.2004  10.223  < 2e-16 ***
## x1            1.2792     0.1720   7.438 6.98e-12 ***
## x4            1.9442     0.3506   5.545 1.27e-07 ***
## x2            1.1886     0.2369   5.017 1.45e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.558 on 152 degrees of freedom
## Multiple R-squared:  0.7536, Adjusted R-squared:  0.7487 
## F-statistic: 154.9 on 3 and 152 DF,  p-value: < 2.2e-16
add1(mod3,~.+x3+x5+x6, test='F')
## Single term additions
## 
## Model:
## y ~ x1 + x4 + x2
##        Df Sum of Sq    RSS     AIC F value    Pr(>F)    
## <none>              47.329 -178.07                      
## x3      1    3.3363 43.993 -187.47 11.4514 0.0009103 ***
## x5      1    0.7866 46.542 -178.68  2.5522 0.1122326    
## x6      1    1.6519 45.677 -181.61  5.4608 0.0207613 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
mod4 = update(mod3,~.+x3)
summary(mod4)
## 
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.86584 -0.34594  0.03403  0.43676  1.13076 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   1.8921     0.1994   9.491  < 2e-16 ***
## x1            0.8105     0.2165   3.745 0.000256 ***
## x4            1.8458     0.3404   5.423 2.28e-07 ***
## x2            1.0166     0.2347   4.331 2.70e-05 ***
## x3            1.1414     0.3373   3.384 0.000910 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5398 on 151 degrees of freedom
## Multiple R-squared:  0.7709, Adjusted R-squared:  0.7649 
## F-statistic:   127 on 4 and 151 DF,  p-value: < 2.2e-16
add1(mod4,~.+x5+x6, test='F')
## Single term additions
## 
## Model:
## y ~ x1 + x4 + x2 + x3
##        Df Sum of Sq    RSS     AIC F value  Pr(>F)  
## <none>              43.993 -187.47                  
## x5      1    0.6661 43.326 -187.85  2.3061 0.13097  
## x6      1    1.3053 42.687 -190.17  4.5866 0.03384 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
mod5 = update(mod4,~.+x6)
summary(mod5)
## 
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x6)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.82997 -0.35344  0.05803  0.35977  1.17522 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   1.8689     0.1973   9.471  < 2e-16 ***
## x1            0.7455     0.2161   3.450 0.000728 ***
## x4            1.5340     0.3666   4.185 4.84e-05 ***
## x2            1.1180     0.2368   4.722 5.33e-06 ***
## x3            1.0840     0.3344   3.241 0.001467 ** 
## x6            1.1176     0.5218   2.142 0.033839 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5335 on 150 degrees of freedom
## Multiple R-squared:  0.7777, Adjusted R-squared:  0.7703 
## F-statistic:   105 on 5 and 150 DF,  p-value: < 2.2e-16
add1(mod5,~.+x5, test='F')
## Single term additions
## 
## Model:
## y ~ x1 + x4 + x2 + x3 + x6
##        Df Sum of Sq    RSS     AIC F value Pr(>F)
## <none>              42.687 -190.17               
## x5      1   0.27561 42.412 -189.18  0.9683 0.3267
add1(mod5,~.+x1*x2+x1*x3+x1*x4+x1*x5+x1*x6+x2*x3+x2*x4+x2*x5+x2*x6+x3*x4+x3*x5+x3*x6+x4*x5+x4*x6+x5*x6, test='F')
## Single term additions
## 
## Model:
## y ~ x1 + x4 + x2 + x3 + x6
##        Df Sum of Sq    RSS     AIC F value    Pr(>F)    
## <none>              42.687 -190.17                      
## x5      1    0.2756 42.412 -189.18  0.9683 0.3267089    
## x1:x2   1    5.0857 37.602 -207.96 20.1527 1.422e-05 ***
## x1:x3   1    4.2664 38.421 -204.60 16.5456 7.671e-05 ***
## x1:x4   1    1.6220 41.065 -194.21  5.8852 0.0164632 *  
## x1:x6   1    1.2775 41.410 -192.91  4.5966 0.0336580 *  
## x2:x3   1    7.2998 35.388 -217.43 30.7360 1.310e-07 ***
## x4:x2   1    2.5442 40.143 -197.75  9.4433 0.0025206 ** 
## x2:x6   1    3.7156 38.972 -202.38 14.2056 0.0002356 ***
## x4:x3   1    1.6189 41.068 -194.20  5.8737 0.0165675 *  
## x3:x6   1    0.8436 41.844 -191.28  3.0039 0.0851320 .  
## x4:x6   1    0.3380 42.349 -189.41  1.1891 0.2772766    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
mod6 = update(mod5,~.+x2*x3)
summary(mod6)
## 
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x6 + x2:x3)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.63890 -0.23831  0.02109  0.33968  1.26013 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   3.9233     0.4121   9.520  < 2e-16 ***
## x1            0.6484     0.1982   3.272 0.001326 ** 
## x4            1.6121     0.3352   4.810 3.67e-06 ***
## x2           -0.6932     0.3918  -1.769 0.078927 .  
## x3           -2.6326     0.7367  -3.573 0.000475 ***
## x6            0.2696     0.5006   0.538 0.591068    
## x2:x3         3.2111     0.5792   5.544 1.31e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4873 on 149 degrees of freedom
## Multiple R-squared:  0.8157, Adjusted R-squared:  0.8083 
## F-statistic: 109.9 on 6 and 149 DF,  p-value: < 2.2e-16
mod7 = update(mod6,~.-x6)
summary(mod7)
## 
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2:x3)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.6411 -0.2478  0.0184  0.3616  1.2651 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   3.9894     0.3925  10.164  < 2e-16 ***
## x1            0.6598     0.1966   3.357 0.001000 ** 
## x4            1.6826     0.3078   5.466 1.87e-07 ***
## x2           -0.7691     0.3647  -2.109 0.036624 *  
## x3           -2.7303     0.7123  -3.833 0.000186 ***
## x2:x3         3.3064     0.5502   6.009 1.35e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4862 on 150 degrees of freedom
## Multiple R-squared:  0.8154, Adjusted R-squared:  0.8092 
## F-statistic: 132.5 on 5 and 150 DF,  p-value: < 2.2e-16
add1(mod7,~.+x1*x2+x1*x3+x1*x4+x1*x5+x1*x6+x2*x4+x2*x5+x2*x6+x3*x4+x3*x5+x3*x6+x4*x5+x4*x6+x5*x6, test='F')
## Single term additions
## 
## Model:
## y ~ x1 + x4 + x2 + x3 + x2:x3
##        Df Sum of Sq    RSS     AIC F value Pr(>F)
## <none>              35.456 -219.12               
## x5      1  0.034508 35.422 -217.27  0.1452 0.7038
## x6      1  0.068859 35.388 -217.43  0.2899 0.5911
## x1:x2   1  0.030737 35.426 -217.26  0.1293 0.7197
## x1:x3   1  0.087048 35.369 -217.51  0.3667 0.5457
## x1:x4   1  0.083232 35.373 -217.49  0.3506 0.5547
## x4:x2   1  0.105871 35.351 -217.59  0.4462 0.5052
## x4:x3   1  0.004253 35.452 -217.14  0.0179 0.8938
#summary(mod8)
# b)AIC:
mod.i = lm(y~(x1+x2+x3+x4+x5+x6)^2)
step(mod0,scope = list(lower=mod0,upper=mod.i))
## Start:  AIC=34.43
## y ~ 1
## 
##        Df Sum of Sq     RSS      AIC
## + x1    1   121.040  71.011 -118.776
## + x3    1   116.809  75.242 -109.747
## + x2    1   115.964  76.087 -108.005
## + x4    1    61.686 130.365  -24.005
## + x6    1    28.557 163.493   11.319
## <none>              192.051   34.433
## + x5    1     1.104 190.946   35.533
## 
## Step:  AIC=-118.78
## y ~ x1
## 
##        Df Sum of Sq     RSS      AIC
## + x4    1    15.845  55.166 -156.164
## + x2    1    14.108  56.903 -151.327
## + x3    1     8.649  62.361 -137.038
## + x6    1     4.639  66.372 -127.314
## + x5    1     3.738  67.273 -125.212
## <none>               71.011 -118.776
## - x1    1   121.040 192.051   34.433
## 
## Step:  AIC=-156.16
## y ~ x1 + x4
## 
##         Df Sum of Sq     RSS      AIC
## + x2     1     7.837  47.329 -178.067
## + x3     1     5.709  49.457 -171.207
## + x1:x4  1     0.901  54.265 -156.733
## <none>                55.166 -156.164
## + x6     1     0.545  54.621 -155.713
## + x5     1     0.457  54.709 -155.461
## - x4     1    15.845  71.011 -118.776
## - x1     1    75.199 130.365  -24.005
## 
## Step:  AIC=-178.07
## y ~ x1 + x4 + x2
## 
##         Df Sum of Sq    RSS     AIC
## + x1:x2  1    6.5207 40.808 -199.19
## + x2:x4  1    3.4675 43.861 -187.94
## + x3     1    3.3363 43.993 -187.47
## + x1:x4  1    2.0414 45.287 -182.94
## + x6     1    1.6519 45.677 -181.61
## + x5     1    0.7866 46.542 -178.68
## <none>               47.329 -178.07
## - x2     1    7.8369 55.166 -156.16
## - x4     1    9.5743 56.903 -151.33
## - x1     1   17.2282 64.557 -131.64
## 
## Step:  AIC=-199.19
## y ~ x1 + x4 + x2 + x1:x2
## 
##         Df Sum of Sq    RSS     AIC
## + x3     1    3.1524 37.656 -209.73
## + x2:x4  1    0.7128 40.095 -199.94
## <none>               40.808 -199.19
## + x5     1    0.1475 40.661 -197.76
## + x6     1    0.1377 40.670 -197.72
## + x1:x4  1    0.0731 40.735 -197.47
## - x1:x2  1    6.5207 47.329 -178.07
## - x4     1    7.4742 48.282 -174.96
## 
## Step:  AIC=-209.73
## y ~ x1 + x4 + x2 + x3 + x1:x2
## 
##         Df Sum of Sq    RSS     AIC
## + x2:x3  1    2.2301 35.426 -217.26
## + x2:x4  1    0.7140 36.942 -210.72
## + x1:x3  1    0.5242 37.132 -209.92
## <none>               37.656 -209.73
## + x3:x4  1    0.2800 37.376 -208.90
## + x1:x4  1    0.2102 37.446 -208.61
## + x5     1    0.1029 37.553 -208.16
## + x6     1    0.0542 37.602 -207.96
## - x3     1    3.1524 40.808 -199.19
## - x1:x2  1    6.3368 43.993 -187.47
## - x4     1    6.6508 44.307 -186.36
## 
## Step:  AIC=-217.26
## y ~ x1 + x4 + x2 + x3 + x1:x2 + x2:x3
## 
##         Df Sum of Sq    RSS     AIC
## - x1:x2  1    0.0307 35.456 -219.12
## <none>               35.426 -217.26
## + x1:x3  1    0.1298 35.296 -215.83
## + x1:x4  1    0.1027 35.323 -215.71
## + x6     1    0.0953 35.330 -215.68
## + x2:x4  1    0.0936 35.332 -215.67
## + x5     1    0.0354 35.390 -215.41
## + x3:x4  1    0.0031 35.423 -215.27
## - x2:x3  1    2.2301 37.656 -209.73
## - x4     1    7.0798 42.505 -190.84
## 
## Step:  AIC=-219.12
## y ~ x1 + x4 + x2 + x3 + x2:x3
## 
##         Df Sum of Sq    RSS     AIC
## <none>               35.456 -219.12
## + x2:x4  1    0.1059 35.351 -217.59
## + x1:x3  1    0.0870 35.369 -217.51
## + x1:x4  1    0.0832 35.373 -217.49
## + x6     1    0.0689 35.388 -217.43
## + x5     1    0.0345 35.422 -217.27
## + x1:x2  1    0.0307 35.426 -217.26
## + x3:x4  1    0.0043 35.452 -217.14
## - x1     1    2.6630 38.119 -209.82
## - x4     1    7.0631 42.519 -192.78
## - x2:x3  1    8.5362 43.993 -187.47
## 
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2:x3)
## 
## Coefficients:
## (Intercept)           x1           x4           x2           x3        x2:x3  
##      3.9894       0.6598       1.6826      -0.7691      -2.7303       3.3064
# c)BIC:
step(mod0,scope = list(lower=mod0,upper=mod.i), k=log(n))
## Start:  AIC=37.48
## y ~ 1
## 
##        Df Sum of Sq     RSS      AIC
## + x1    1   121.040  71.011 -112.676
## + x3    1   116.809  75.242 -103.647
## + x2    1   115.964  76.087 -101.905
## + x4    1    61.686 130.365  -17.906
## + x6    1    28.557 163.493   17.418
## <none>              192.051   37.483
## + x5    1     1.104 190.946   41.633
## 
## Step:  AIC=-112.68
## y ~ x1
## 
##        Df Sum of Sq     RSS      AIC
## + x4    1    15.845  55.166 -147.015
## + x2    1    14.108  56.903 -142.177
## + x3    1     8.649  62.361 -127.888
## + x6    1     4.639  66.372 -118.165
## + x5    1     3.738  67.273 -116.062
## <none>               71.011 -112.676
## - x1    1   121.040 192.051   37.483
## 
## Step:  AIC=-147.01
## y ~ x1 + x4
## 
##         Df Sum of Sq     RSS      AIC
## + x2     1     7.837  47.329 -165.867
## + x3     1     5.709  49.457 -159.007
## <none>                55.166 -147.015
## + x1:x4  1     0.901  54.265 -144.534
## + x6     1     0.545  54.621 -143.514
## + x5     1     0.457  54.709 -143.261
## - x4     1    15.845  71.011 -112.676
## - x1     1    75.199 130.365  -17.906
## 
## Step:  AIC=-165.87
## y ~ x1 + x4 + x2
## 
##         Df Sum of Sq    RSS     AIC
## + x1:x2  1    6.5207 40.808 -183.94
## + x2:x4  1    3.4675 43.861 -172.69
## + x3     1    3.3363 43.993 -172.22
## + x1:x4  1    2.0414 45.287 -167.70
## + x6     1    1.6519 45.677 -166.36
## <none>               47.329 -165.87
## + x5     1    0.7866 46.542 -163.43
## - x2     1    7.8369 55.166 -147.01
## - x4     1    9.5743 56.903 -142.18
## - x1     1   17.2282 64.557 -122.49
## 
## Step:  AIC=-183.94
## y ~ x1 + x4 + x2 + x1:x2
## 
##         Df Sum of Sq    RSS     AIC
## + x3     1    3.1524 37.656 -191.44
## <none>               40.808 -183.94
## + x2:x4  1    0.7128 40.095 -181.64
## + x5     1    0.1475 40.661 -179.46
## + x6     1    0.1377 40.670 -179.42
## + x1:x4  1    0.0731 40.735 -179.17
## - x1:x2  1    6.5207 47.329 -165.87
## - x4     1    7.4742 48.282 -162.76
## 
## Step:  AIC=-191.43
## y ~ x1 + x4 + x2 + x3 + x1:x2
## 
##         Df Sum of Sq    RSS     AIC
## + x2:x3  1    2.2301 35.426 -195.91
## <none>               37.656 -191.44
## + x2:x4  1    0.7140 36.942 -189.37
## + x1:x3  1    0.5242 37.132 -188.57
## + x3:x4  1    0.2800 37.376 -187.55
## + x1:x4  1    0.2102 37.446 -187.26
## + x5     1    0.1029 37.553 -186.81
## + x6     1    0.0542 37.602 -186.61
## - x3     1    3.1524 40.808 -183.94
## - x1:x2  1    6.3368 43.993 -172.22
## - x4     1    6.6508 44.307 -171.11
## 
## Step:  AIC=-195.91
## y ~ x1 + x4 + x2 + x3 + x1:x2 + x2:x3
## 
##         Df Sum of Sq    RSS     AIC
## - x1:x2  1    0.0307 35.456 -200.82
## <none>               35.426 -195.91
## - x2:x3  1    2.2301 37.656 -191.44
## + x1:x3  1    0.1298 35.296 -191.43
## + x1:x4  1    0.1027 35.323 -191.31
## + x6     1    0.0953 35.330 -191.28
## + x2:x4  1    0.0936 35.332 -191.27
## + x5     1    0.0354 35.390 -191.01
## + x3:x4  1    0.0031 35.423 -190.87
## - x4     1    7.0798 42.505 -172.54
## 
## Step:  AIC=-200.82
## y ~ x1 + x4 + x2 + x3 + x2:x3
## 
##         Df Sum of Sq    RSS     AIC
## <none>               35.456 -200.82
## + x2:x4  1    0.1059 35.351 -196.24
## + x1:x3  1    0.0870 35.369 -196.16
## + x1:x4  1    0.0832 35.373 -196.14
## + x6     1    0.0689 35.388 -196.08
## + x5     1    0.0345 35.422 -195.93
## + x1:x2  1    0.0307 35.426 -195.91
## + x3:x4  1    0.0043 35.452 -195.79
## - x1     1    2.6630 38.119 -194.57
## - x4     1    7.0631 42.519 -177.53
## - x2:x3  1    8.5362 43.993 -172.22
## 
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2:x3)
## 
## Coefficients:
## (Intercept)           x1           x4           x2           x3        x2:x3  
##      3.9894       0.6598       1.6826      -0.7691      -2.7303       3.3064
# NOT RUN {
AIC(mod0)
## [1] 479.1416
AIC(mod1)
## [1] 325.9328
AIC(mod2)
## [1] 288.5446
AIC(mod3)
## [1] 266.6419
AIC(mod4)
## [1] 257.2385
AIC(mod5)
## [1] 254.54
AIC(mod6)
## [1] 227.2833
AIC(mod7)
## [1] 225.5866
#stopifnot(all.equal(AIC(mod0), AIC(logLik(mod0))))
BIC(mod7)
## [1] 246.9356
#lm2 <- update(lm1, . ~ . -Examination)
#AIC(lm1, lm2)
#BIC(lm1, lm2)
# }
# Model after Step-wise regression:
mod8 = lm(y~x1+x4+x2+x3+x2*x3)
mod = regsubsets(y~(x1+x2+x3+x4+x5+x6)^2,data=dat,nvmax=8)
summary(mod)
## Subset selection object
## Call: regsubsets.formula(y ~ (x1 + x2 + x3 + x4 + x5 + x6)^2, data = dat, 
##     nvmax = 8)
## 21 Variables  (and intercept)
##       Forced in Forced out
## x1        FALSE      FALSE
## x2        FALSE      FALSE
## x3        FALSE      FALSE
## x4        FALSE      FALSE
## x5        FALSE      FALSE
## x6        FALSE      FALSE
## x1:x2     FALSE      FALSE
## x1:x3     FALSE      FALSE
## x1:x4     FALSE      FALSE
## x1:x5     FALSE      FALSE
## x1:x6     FALSE      FALSE
## x2:x3     FALSE      FALSE
## x2:x4     FALSE      FALSE
## x2:x5     FALSE      FALSE
## x2:x6     FALSE      FALSE
## x3:x4     FALSE      FALSE
## x3:x5     FALSE      FALSE
## x3:x6     FALSE      FALSE
## x4:x5     FALSE      FALSE
## x4:x6     FALSE      FALSE
## x5:x6     FALSE      FALSE
## 1 subsets of each size up to 8
## Selection Algorithm: exhaustive
##          x1  x2  x3  x4  x5  x6  x1:x2 x1:x3 x1:x4 x1:x5 x1:x6 x2:x3 x2:x4
## 1  ( 1 ) " " " " " " " " " " " " " "   " "   " "   " "   " "   "*"   " "  
## 2  ( 1 ) " " " " " " " " " " " " " "   " "   "*"   " "   " "   "*"   " "  
## 3  ( 1 ) " " " " "*" " " " " " " " "   " "   "*"   " "   " "   "*"   " "  
## 4  ( 1 ) " " " " " " " " " " " " " "   " "   "*"   " "   " "   "*"   " "  
## 5  ( 1 ) " " "*" " " " " " " " " " "   " "   "*"   " "   " "   "*"   " "  
## 6  ( 1 ) " " " " " " "*" " " " " " "   " "   " "   "*"   " "   "*"   " "  
## 7  ( 1 ) " " "*" " " "*" " " " " " "   " "   " "   "*"   " "   "*"   " "  
## 8  ( 1 ) " " "*" "*" "*" " " " " " "   " "   " "   "*"   " "   "*"   " "  
##          x2:x5 x2:x6 x3:x4 x3:x5 x3:x6 x4:x5 x4:x6 x5:x6
## 1  ( 1 ) " "   " "   " "   " "   " "   " "   " "   " "  
## 2  ( 1 ) " "   " "   " "   " "   " "   " "   " "   " "  
## 3  ( 1 ) " "   " "   " "   " "   " "   " "   " "   " "  
## 4  ( 1 ) " "   "*"   " "   " "   "*"   " "   " "   " "  
## 5  ( 1 ) " "   "*"   " "   " "   "*"   " "   " "   " "  
## 6  ( 1 ) " "   "*"   " "   " "   "*"   "*"   " "   " "  
## 7  ( 1 ) " "   "*"   " "   " "   "*"   "*"   " "   " "  
## 8  ( 1 ) " "   "*"   " "   " "   "*"   "*"   " "   " "
sm = summary(mod)
sm$which
##   (Intercept)    x1    x2    x3    x4    x5    x6 x1:x2 x1:x3 x1:x4 x1:x5 x1:x6
## 1        TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## 2        TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE FALSE
## 3        TRUE FALSE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE FALSE
## 4        TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE FALSE
## 5        TRUE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE FALSE
## 6        TRUE FALSE FALSE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE
## 7        TRUE FALSE  TRUE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE
## 8        TRUE FALSE  TRUE  TRUE  TRUE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE
##   x2:x3 x2:x4 x2:x5 x2:x6 x3:x4 x3:x5 x3:x6 x4:x5 x4:x6 x5:x6
## 1  TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## 2  TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## 3  TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## 4  TRUE FALSE FALSE  TRUE FALSE FALSE  TRUE FALSE FALSE FALSE
## 5  TRUE FALSE FALSE  TRUE FALSE FALSE  TRUE FALSE FALSE FALSE
## 6  TRUE FALSE FALSE  TRUE FALSE FALSE  TRUE  TRUE FALSE FALSE
## 7  TRUE FALSE FALSE  TRUE FALSE FALSE  TRUE  TRUE FALSE FALSE
## 8  TRUE FALSE FALSE  TRUE FALSE FALSE  TRUE  TRUE FALSE FALSE
sm$adjr2
## [1] 0.7448183 0.7970237 0.8067916 0.8124593 0.8147822 0.8161633 0.8202197
## [8] 0.8217300
sm$adjr2[6]
## [1] 0.8161633
sm$adjr2[8]
## [1] 0.82173
rss = sm$rss
mses = c(rss[1]/(n-2), rss[2]/(n-3), rss[3]/(n-4), rss[4]/(n-5), rss[5]/(n-6), rss[6]/(n-7), rss[7]/(n-8), rss[8]/(n-9))
mses
## [1] 0.3161793 0.2514948 0.2393921 0.2323697 0.2294915 0.2277803 0.2227543
## [8] 0.2208829
sm$cp
## [1] 61.943043 19.069203 11.881227  8.170178  7.252268  7.123605  4.854404
## [8]  4.666938
sm$cp[6]
## [1] 7.123605
mod.sub = lm(y~x1+x4+x2+x3+x5+x6+x1*x5+x2*x3+x2*x6+x3*x6+x4*x5) 
plot(mod.sub)

hist(resid(mod.sub))

# Model Assumptions
summary(mod8)
## 
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2 * x3)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.6411 -0.2478  0.0184  0.3616  1.2651 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   3.9894     0.3925  10.164  < 2e-16 ***
## x1            0.6598     0.1966   3.357 0.001000 ** 
## x4            1.6826     0.3078   5.466 1.87e-07 ***
## x2           -0.7691     0.3647  -2.109 0.036624 *  
## x3           -2.7303     0.7123  -3.833 0.000186 ***
## x2:x3         3.3064     0.5502   6.009 1.35e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4862 on 150 degrees of freedom
## Multiple R-squared:  0.8154, Adjusted R-squared:  0.8092 
## F-statistic: 132.5 on 5 and 150 DF,  p-value: < 2.2e-16
plot(mod8)

#Model Assumptions
summary(mod8)
## 
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2 * x3)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.6411 -0.2478  0.0184  0.3616  1.2651 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   3.9894     0.3925  10.164  < 2e-16 ***
## x1            0.6598     0.1966   3.357 0.001000 ** 
## x4            1.6826     0.3078   5.466 1.87e-07 ***
## x2           -0.7691     0.3647  -2.109 0.036624 *  
## x3           -2.7303     0.7123  -3.833 0.000186 ***
## x2:x3         3.3064     0.5502   6.009 1.35e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4862 on 150 degrees of freedom
## Multiple R-squared:  0.8154, Adjusted R-squared:  0.8092 
## F-statistic: 132.5 on 5 and 150 DF,  p-value: < 2.2e-16
hist(resid(mod8))

shapiro.test(resid(mod8))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(mod8)
## W = 0.9832, p-value = 0.05482
dat2 = dat[-c(155),]
y = dat2$Score
x1 = dat2$GDP.per.capita
x2 = dat2$Social.support
x3 = dat2$Healthy.life.expectancy
x4 = dat2$Freedom.to.make.life.choices 
x5 = dat2$Generosity
x6 = dat2$Perceptions.of.corruption 
mod9 = lm(y~x1+x4+x2+x3+x2*x3) 
summary(mod9)
## 
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2 * x3)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.60976 -0.22497  0.02944  0.30369  1.24555 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   4.9521     0.5093   9.724  < 2e-16 ***
## x1            0.6810     0.1921   3.544 0.000526 ***
## x4            1.6863     0.3006   5.609 9.61e-08 ***
## x2           -1.6227     0.4639  -3.498 0.000619 ***
## x3           -4.1455     0.8526  -4.862 2.92e-06 ***
## x2:x3         4.4766     0.6744   6.638 5.56e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4749 on 149 degrees of freedom
## Multiple R-squared:   0.82,  Adjusted R-squared:  0.8139 
## F-statistic: 135.7 on 5 and 149 DF,  p-value: < 2.2e-16
plot(mod9)

hist(resid(mod9))

shapiro.test(resid(mod9))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(mod9)
## W = 0.9791, p-value = 0.01868
boxc = boxcox(y~x1+x4+x2+x3+x2*x3, data = dat2, lambda = seq(-2, 2, 0.1))

lambda = boxc$x[which.max(boxc$y)]
lambda
## [1] 1.434343
mod10 = lm(y^lambda~x1+x4+x2+x3+x2*x3) 
shapiro.test(resid(mod10))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(mod10)
## W = 0.98361, p-value = 0.06295
hist(resid(mod10))

plot(mod10)

# Begin full vs reduced comparison
modred <- lm(y~x1+x2+x3+x4+x2*x3) 
modfull <- lm(y~(x1+x2+x3+x4+x5+x6)^2) 
anova(modred,modfull)
## Analysis of Variance Table
## 
## Model 1: y ~ x1 + x2 + x3 + x4 + x2 * x3
## Model 2: y ~ (x1 + x2 + x3 + x4 + x5 + x6)^2
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1    149 33.597                           
## 2    133 28.564 16     5.033 1.4646 0.1222
modred2 <- mod10
anova(modred2,modfull)
## Warning in anova.lmlist(object, ...): models with response '"y"' removed because
## response differs from model 1
## Analysis of Variance Table
## 
## Response: y^lambda
##            Df  Sum Sq Mean Sq F value    Pr(>F)    
## x1          1 1024.23 1024.23 527.991 < 2.2e-16 ***
## x4          1  142.49  142.49  73.454 1.209e-14 ***
## x2          1   66.24   66.24  34.148 3.099e-08 ***
## x3          1   30.79   30.79  15.873 0.0001057 ***
## x2:x3       1  110.17  110.17  56.793 4.353e-12 ***
## Residuals 149  289.04    1.94                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Model after Step-wise regression:
mod8 = lm(y~x1+x4+x2+x3+x2*x3)

# Best Subsets Regression
mod = regsubsets(cbind(x1,x2,x3,x4,x5,x6),y)
sm = summary(mod)
sm$which
##   (Intercept)   x1    x2    x3    x4    x5    x6
## 1        TRUE TRUE FALSE FALSE FALSE FALSE FALSE
## 2        TRUE TRUE FALSE FALSE  TRUE FALSE FALSE
## 3        TRUE TRUE  TRUE FALSE  TRUE FALSE FALSE
## 4        TRUE TRUE  TRUE  TRUE  TRUE FALSE FALSE
## 5        TRUE TRUE  TRUE  TRUE  TRUE FALSE  TRUE
## 6        TRUE TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
sm$adjr2
## [1] 0.6177759 0.7008756 0.7433231 0.7605097 0.7666651 0.7664649
rss = sm$rss
mses = c(rss[1]/(n-2), rss[2]/(n-3), rss[3]/(n-4), rss[4]/(n-5), rss[5]/(n-6), rss[6]/(n-7)) 
mses
## [1] 0.4601643 0.3601042 0.3089899 0.2882878 0.2808658 0.2810941
sm$cp
## [1] 99.413265 45.689832 18.963116  8.825016  5.872250  7.000000
# Model after Best Subsets Regression
mod.sub = lm(y~x1+x4+x2+x3+x6)

# Model Assumptions
summary(mod8)
## 
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2 * x3)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.60976 -0.22497  0.02944  0.30369  1.24555 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   4.9521     0.5093   9.724  < 2e-16 ***
## x1            0.6810     0.1921   3.544 0.000526 ***
## x4            1.6863     0.3006   5.609 9.61e-08 ***
## x2           -1.6227     0.4639  -3.498 0.000619 ***
## x3           -4.1455     0.8526  -4.862 2.92e-06 ***
## x2:x3         4.4766     0.6744   6.638 5.56e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4749 on 149 degrees of freedom
## Multiple R-squared:   0.82,  Adjusted R-squared:  0.8139 
## F-statistic: 135.7 on 5 and 149 DF,  p-value: < 2.2e-16
plot(mod8)

hist(resid(mod8))

shapiro.test(resid(mod8))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(mod8)
## W = 0.9791, p-value = 0.01868
dat2 = dat[-c(155),]
y = dat2$Score
x1 = dat2$GDP.per.capita
x2 = dat2$Social.support
x3 = dat2$Healthy.life.expectancy
x4 = dat2$Freedom.to.make.life.choices
x5 = dat2$Generosity
x6 = dat2$Perceptions.of.corruption
mod9 = lm(y~x1+x4+x2+x3+x2*x3)
summary(mod9)
## 
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2 * x3)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.60976 -0.22497  0.02944  0.30369  1.24555 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   4.9521     0.5093   9.724  < 2e-16 ***
## x1            0.6810     0.1921   3.544 0.000526 ***
## x4            1.6863     0.3006   5.609 9.61e-08 ***
## x2           -1.6227     0.4639  -3.498 0.000619 ***
## x3           -4.1455     0.8526  -4.862 2.92e-06 ***
## x2:x3         4.4766     0.6744   6.638 5.56e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4749 on 149 degrees of freedom
## Multiple R-squared:   0.82,  Adjusted R-squared:  0.8139 
## F-statistic: 135.7 on 5 and 149 DF,  p-value: < 2.2e-16
plot(mod9)

shapiro.test(resid(mod9))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(mod9)
## W = 0.9791, p-value = 0.01868
boxc = boxcox(y~x1+x4+x2+x3+x2*x3, data = dat2, lambda = seq(-2, 2, 0.1))

lambda = boxc$x[which.max(boxc$y)]
mod10 = lm(y^lambda~x1+x4+x2+x3+x2*x3)
shapiro.test(resid(mod10))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(mod10)
## W = 0.98361, p-value = 0.06295
plot(mod10)

# Begin full vs reduced comparison
modred <- lm(y~x1+x2+x3+x4+x2*x3)
modfull <- lm(y~(x1+x2+x3+x4+x5+x6+I(x6^2))^2)
anova(modred,modfull)
## Analysis of Variance Table
## 
## Model 1: y ~ x1 + x2 + x3 + x4 + x2 * x3
## Model 2: y ~ (x1 + x2 + x3 + x4 + x5 + x6 + I(x6^2))^2
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1    149 33.597                           
## 2    126 26.567 23    7.0306 1.4498 0.1009
# Conclude reduced is better
summary(mod9)
## 
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2 * x3)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.60976 -0.22497  0.02944  0.30369  1.24555 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   4.9521     0.5093   9.724  < 2e-16 ***
## x1            0.6810     0.1921   3.544 0.000526 ***
## x4            1.6863     0.3006   5.609 9.61e-08 ***
## x2           -1.6227     0.4639  -3.498 0.000619 ***
## x3           -4.1455     0.8526  -4.862 2.92e-06 ***
## x2:x3         4.4766     0.6744   6.638 5.56e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4749 on 149 degrees of freedom
## Multiple R-squared:   0.82,  Adjusted R-squared:  0.8139 
## F-statistic: 135.7 on 5 and 149 DF,  p-value: < 2.2e-16
cooks.distance(mod7)
##            1            2            3            4            5            6 
## 1.049689e-02 5.170036e-03 1.432856e-03 6.995258e-04 5.120677e-03 2.407868e-03 
##            7            8            9           10           11           12 
## 3.221363e-03 9.796633e-04 1.200415e-03 2.777868e-03 3.148469e-04 9.753133e-03 
##           13           14           15           16           17           18 
## 8.193624e-03 2.948024e-04 1.325853e-03 2.568323e-05 1.428595e-03 5.407540e-04 
##           19           20           21           22           23           24 
## 4.087471e-03 1.631698e-03 1.265441e-02 7.375629e-04 3.874418e-03 2.029278e-04 
##           25           26           27           28           29           30 
## 8.252020e-04 2.278081e-03 1.151198e-02 3.195050e-03 1.799134e-04 1.789754e-03 
##           31           32           33           34           35           36 
## 1.586001e-05 2.074131e-03 5.996229e-05 3.058460e-02 8.424565e-03 7.534879e-04 
##           37           38           39           40           41           42 
## 1.743230e-04 4.888054e-07 5.936122e-04 8.233536e-05 9.565630e-05 4.943048e-04 
##           43           44           45           46           47           48 
## 2.110463e-04 7.739605e-03 7.194322e-03 3.956601e-03 5.453470e-05 1.757552e-03 
##           49           50           51           52           53           54 
## 2.986060e-04 5.479707e-04 3.072500e-04 1.763946e-04 3.473707e-04 6.320577e-03 
##           55           56           57           58           59           60 
## 4.475588e-03 3.395319e-04 1.268331e-04 1.188379e-02 4.935722e-03 2.818429e-04 
##           61           62           63           64           65           66 
## 2.652332e-03 4.013287e-04 7.617066e-04 2.008236e-03 4.582002e-06 9.894691e-03 
##           67           68           69           70           71           72 
## 2.094039e-02 8.913423e-05 3.905539e-04 1.573656e-05 5.413600e-03 9.058306e-05 
##           73           74           75           76           77           78 
## 4.592831e-06 1.177645e-02 4.326127e-04 2.589701e-02 1.958027e-03 1.467337e-03 
##           79           80           81           82           83           84 
## 2.715219e-04 2.997218e-03 2.103827e-03 2.719602e-03 1.499058e-03 4.984239e-04 
##           85           86           87           88           89           90 
## 2.892667e-02 3.557102e-03 5.185894e-03 7.380709e-03 2.932327e-02 3.366936e-07 
##           91           92           93           94           95           96 
## 6.583537e-05 2.402727e-04 3.540444e-03 1.042865e-02 4.942719e-04 9.917056e-03 
##           97           98           99          100          101          102 
## 1.467298e-02 4.435309e-03 1.804694e-02 1.111516e-04 1.801153e-03 4.371052e-02 
##          103          104          105          106          107          108 
## 3.276406e-03 5.728701e-04 9.504668e-04 4.376866e-03 1.603222e-04 1.577864e-02 
##          109          110          111          112          113          114 
## 6.498007e-03 1.390536e-04 6.962798e-04 1.101898e-02 4.022505e-03 1.156969e-02 
##          115          116          117          118          119          120 
## 5.078605e-03 1.598727e-03 3.398840e-04 2.187656e-03 6.018901e-03 1.005339e-03 
##          121          122          123          124          125          126 
## 1.188549e-04 5.552154e-03 1.279468e-04 5.047628e-04 4.065722e-03 1.148529e-03 
##          127          128          129          130          131          132 
## 8.611686e-03 7.412165e-04 3.146845e-03 1.431296e-02 9.767987e-03 1.111701e-02 
##          133          134          135          136          137          138 
## 1.978575e-02 1.068321e-04 3.930002e-04 4.551653e-04 4.128009e-03 4.276355e-03 
##          139          140          141          142          143          144 
## 1.309776e-03 1.450567e-02 4.864694e-04 2.035824e-03 9.571560e-05 1.055017e-02 
##          145          146          147          148          149          150 
## 2.412243e-03 1.105206e-02 3.078043e-04 8.366687e-02 2.301207e-05 1.675004e-02 
##          151          152          153          154          155          156 
## 1.981495e-02 6.514437e-02 2.059902e-02 4.258839e-03 1.052905e+00 3.145728e-02
rstudent(mod7)
##           1           2           3           4           5           6 
##  1.32542081  0.94336423  0.44926717  0.29550739  1.06599253  0.64343611 
##           7           8           9          10          11          12 
##  0.85428073  0.39776172  0.47801344  0.84605808  0.23508987  1.47701345 
##          13          14          15          16          17          18 
##  1.38259631  0.22053300  0.55142529 -0.06947194  0.67328962  0.39099678 
##          19          20          21          22          23          24 
##  1.01908911  0.80439196  1.11208292 -0.40367442  1.60561686 -0.21867099 
##          25          26          27          28          29          30 
##  0.49155886  1.01388509  1.88791968  0.76907638 -0.11447083 -0.54912230 
##          31          32          33          34          35          36 
## -0.08220691  0.97811450 -0.15380530 -2.07942901  1.84985134 -0.29331932 
##          37          38          39          40          41          42 
## -0.20820005  0.01155022  0.35010075 -0.20481216  0.10403444  0.32069240 
##          43          44          45          46          47          48 
##  0.31927473 -1.41138591  1.18465802  1.31882261 -0.16697198  0.77895275 
##          49          50          51          52          53          54 
##  0.23370099  0.43961593 -0.19095502 -0.25719221  0.27785986  0.78352594 
##          55          56          57          58          59          60 
## -1.24618086 -0.26521520 -0.24842652 -1.61498213  0.88249219 -0.26726924 
##          61          62          63          64          65          66 
##  0.93981716  0.26342310 -0.43362764 -0.64943582  0.04699062 -1.78873130 
##          67          68          69          70          71          72 
##  2.67930272 -0.14619395  0.31071952 -0.06855319  0.98187648  0.17985252 
##          73          74          75          76          77          78 
##  0.02918037  1.46068161 -0.36088505 -1.99185012 -0.98168783  0.56053802 
##          79          80          81          82          83          84 
## -0.22156122 -0.74150386 -0.63918041  0.43579213 -0.50329621 -0.50892306 
##          85          86          87          88          89          90 
##  1.54767141 -0.64063912 -0.96935141  0.92838063  1.50437350  0.01078424 
##          91          92          93          94          95          96 
##  0.12694374 -0.28099046 -0.77791856 -1.35413392 -0.38483889  1.38071524 
##          97          98          99         100         101         102 
## -1.91214083  1.16881557  1.32179528 -0.13021248 -0.85651344  1.92008584 
##         103         104         105         106         107         108 
##  0.85808121 -0.31043968 -0.41235851 -0.69720503  0.10330213 -1.43097377 
##         109         110         111         112         113         114 
## -0.95293618 -0.17120309  0.40214873  0.83435454 -0.76233404  1.40760747 
##         115         116         117         118         119         120 
##  0.98794881 -0.56994523 -0.15961107  0.76597840  0.58725806  0.50354755 
##         121         122         123         124         125         126 
## -0.18525226  0.75890678  0.13097828 -0.24677967 -0.67754214 -0.38813607 
##         127         128         129         130         131         132 
##  0.90292663  0.33828494  0.67558651 -2.69149175 -1.51632270  0.98638411 
##         133         134         135         136         137         138 
## -1.70574207 -0.15364272  0.10122431 -0.31705080 -1.01224924 -1.04565708 
##         139         140         141         142         143         144 
##  0.42951073 -1.27778413 -0.24553825  0.54619093  0.11052160 -0.86797787 
##         145         146         147         148         149         150 
##  0.43565528 -1.53203749  0.17796384 -3.57911689 -0.03376150 -1.24438600 
##         151         152         153         154         155         156 
## -1.40507600 -2.26522962 -2.51833924 -0.53757664 -2.87153201 -1.44125372
cooks.distance(modred)
##            1            2            3            4            5            6 
## 9.352437e-03 4.125265e-03 5.306230e-04 5.156334e-05 4.499310e-03 1.507651e-03 
##            7            8            9           10           11           12 
## 2.720219e-03 3.754136e-04 6.721946e-04 2.373646e-03 2.542842e-05 9.779971e-03 
##           13           14           15           16           17           18 
## 7.799401e-03 7.243498e-05 8.799056e-04 2.987501e-04 1.235026e-03 3.288777e-04 
##           19           20           21           22           23           24 
## 4.208541e-03 1.555404e-03 1.460258e-02 1.401958e-03 4.560094e-03 5.449076e-04 
##           25           26           27           28           29           30 
## 8.057834e-04 2.472970e-03 1.437483e-02 3.904167e-03 1.004814e-04 3.153717e-03 
##           31           32           33           34           35           36 
## 2.203105e-05 2.509647e-03 7.102555e-05 4.228843e-02 1.070879e-02 1.681624e-03 
##           37           38           39           40           41           42 
## 1.411672e-04 3.949242e-07 1.060533e-03 8.504125e-05 3.450793e-04 5.921786e-04 
##           43           44           45           46           47           48 
## 2.896399e-04 9.522094e-03 8.859869e-03 5.257599e-03 5.030492e-05 2.383619e-03 
##           49           50           51           52           53           54 
## 5.605812e-04 7.835154e-04 1.667473e-04 1.272432e-04 4.761753e-04 8.169628e-03 
##           55           56           57           58           59           60 
## 4.914787e-03 2.638920e-04 8.048048e-05 1.476422e-02 6.766268e-03 1.364433e-04 
##           61           62           63           64           65           66 
## 3.727472e-03 5.694209e-04 5.716173e-04 1.798515e-03 3.641738e-05 1.133607e-02 
##           67           68           69           70           71           72 
## 2.153183e-02 1.363149e-05 8.452966e-04 1.945106e-06 7.365396e-03 2.645710e-04 
##           73           74           75           76           77           78 
## 3.089186e-05 1.522396e-02 3.125018e-04 2.680393e-02 1.859833e-03 2.243804e-03 
##           79           80           81           82           83           84 
## 1.497213e-04 2.471227e-03 1.846719e-03 4.956494e-03 8.434542e-04 4.035726e-04 
##           85           86           87           88           89           90 
## 3.010950e-02 2.549067e-03 4.223515e-03 9.945254e-03 3.983148e-02 4.106783e-05 
##           91           92           93           94           95           96 
## 2.237833e-04 1.142530e-04 2.749101e-03 1.004460e-02 2.394192e-04 8.976609e-03 
##           97           98           99          100          101          102 
## 1.496382e-02 4.185297e-03 1.291458e-02 2.668525e-07 1.603253e-03 3.826041e-02 
##          103          104          105          106          107          108 
## 2.773276e-03 3.234901e-04 8.650037e-04 2.648174e-03 1.952933e-03 1.531348e-02 
##          109          110          111          112          113          114 
## 5.674886e-03 1.446586e-05 1.158194e-03 3.303315e-03 2.808158e-03 1.019760e-02 
##          115          116          117          118          119          120 
## 5.297769e-03 1.050446e-03 2.375946e-06 1.547831e-03 9.882302e-03 8.023214e-04 
##          121          122          123          124          125          126 
## 8.082060e-05 7.545695e-03 4.773599e-05 9.065894e-05 3.062802e-03 1.142957e-03 
##          127          128          129          130          131          132 
## 1.042546e-02 8.634440e-04 1.290339e-03 1.512337e-02 9.404858e-03 4.856524e-03 
##          133          134          135          136          137          138 
## 1.893781e-02 5.937387e-05 4.057156e-04 3.947745e-04 3.866659e-03 4.449472e-03 
##          139          140          141          142          143          144 
## 4.125864e-05 1.642097e-02 8.114311e-04 1.382032e-03 1.131568e-04 9.493885e-03 
##          145          146          147          148          149          150 
## 1.449502e-04 1.090086e-02 2.981156e-05 8.558444e-02 7.707716e-03 2.839985e-02 
##          151          152          153          154          155 
## 1.828324e-02 6.930700e-02 2.324369e-02 2.466200e-02 9.865568e-02
rstudent(modred)
##            1            2            3            4            5            6 
##  1.197388899  0.804773657  0.257762325  0.075134334  0.961812605  0.481962900 
##            7            8            9           10           11           12 
##  0.762043160  0.234351534  0.343031204  0.758332654  0.063183264  1.474110728 
##           13           14           15           16           17           18 
##  1.319122538  0.106675322  0.430783342 -0.226399653  0.614634184  0.295363482 
##           19           20           21           22           23           24 
##  1.033928025  0.778926095  1.191950838 -0.537919547  1.705060609 -0.345945823 
##           25           26           27           28           29           30 
##  0.485240824  1.055280871  2.044463237  0.845137782 -0.085474252 -0.704753526 
##           31           32           33           34           35           36 
## -0.096819077  1.059964874 -0.167326066 -2.334530993  2.003604023 -0.429646069 
##           37           38           39           40           41           42 
## -0.187014756 -0.010367147  0.458325026 -0.208144285  0.195722835  0.350644682 
##           43           44           45           46           47           48 
##  0.370560696 -1.536162450  1.297868594  1.455530280 -0.160264390  0.884554717 
##           49           50           51           52           53           54 
##  0.316621216  0.517357136 -0.140115046 -0.216582629  0.324159420  0.885075888 
##           55           56           57           58           59           60 
## -1.303274044 -0.233044399 -0.194373258 -1.761297395  1.014244317 -0.181903159 
##           61           62           63           64           65           66 
##  1.072535526  0.312744408 -0.370420305 -0.610514619  0.128409366 -1.894857812 
##           67           68           69           70           71           72 
##  2.702513839 -0.055941086  0.440588959 -0.023941369  1.120113435  0.294438883 
##           73           74           75           76           77           78 
##  0.075372932  1.620173635 -0.302428405 -2.026922221 -0.931019372  0.677553637 
##           79           80           81           82           83           84 
## -0.163250487 -0.660777294 -0.594371531  0.580623316 -0.362427508 -0.443821945 
##           85           86           87           88           89           90 
##  1.579524106 -0.532015244 -0.835641787  1.062638952  1.716832437  0.114747803 
##           91           92           93           94           95           96 
##  0.228457366 -0.187364166 -0.664457604 -1.317083565 -0.252311350  1.236199296 
##           97           98           99          100          101          102 
## -1.926642967  1.107093544  0.973386350 -0.006218343 -0.773810371  1.453673003 
##          103          104          105          106          107          108 
##  0.766880790 -0.230098623 -0.392710135 -0.518575676  0.346172187 -1.401771537 
##          109          110          111          112          113          114 
## -0.877757850 -0.053476281  0.507663265  0.399323394 -0.609533351  1.186267361 
##          115          116          117          118          119          120 
##  1.009171177 -0.443466867 -0.013099783  0.577876993  0.743526908  0.443743047 
##          121          122          123          124          125          126 
## -0.152106593  0.875964688  0.079660417 -0.101590962 -0.578551234 -0.387127894 
##          127          128          129          130          131          132 
##  0.989818216  0.364923086  0.377386563 -2.667574662 -1.452494496  0.542968234 
##          133          134          135          136          137          138 
## -1.639594652 -0.114038492  0.102846629 -0.294636275 -0.967738390 -1.066727642 
##          139          140          141          142          143          144 
##  0.063639688 -1.356867781 -0.315440627  0.440206999  0.120160186 -0.820130245 
##          145          146          147          148          149          150 
## -0.086962922 -1.510020662 -0.052252862 -3.597992350 -0.543799886 -1.532878877 
##          151          152          153          154          155 
## -1.333283752 -2.338747141 -2.649260672 -1.089632336 -2.106785029